Derivative of vector-valued functions

 Objective: Define the derivative of a vector-valued function

Since the limit of a vector-valued function has been defined, we can now define its derivative. The definition of the derivative of a vector-valued is similar to that of real-valued functions. The only difference is in the range of the derivative of vector-valued functions. Since the range of a vector-valued function is made of vectors, the range of the derivative of the vector-valued function consists also of vectors. 

Definition

The derivative of a vector-valued is defined by:

provided that the limit exists. If r'(t) exists, then r is differentiable at t. If r is differentiable for all values of t in an open interval (a, b), then r is differentiable in this interval. The following two limits'  must exist as well for the function r to be differentiable in a closed interval [a, b]. 

and

Example

Solution

Let's apply the formula:

Practice

Use the definition to find the derivative of the vector-valued function: